10-direction clock

[Eric Angelini]

                          . 11                

               0
        9      *      1
         *          *

     8 *                * 2
               .                  . 23 & 32

     7 *                * 3

         *          *
        6      *     4
               5

Hello SeqFans,
Imagine we assign a direction to each digit, as above.
If we start at the center of the circle, the number "16"
would bring us back to the center (we move one step in
direction "1" and, from there, one step in direction "6").
The number "11" would drive us outside the clock, as shown,
-- and so would "23" or "32" (sharing the same spot).

What could be a sequence of integers a(n) where all a(n)s
would lead us on _different_ spots? (so if 23 is in the
seq, 32 would not be allowed to).

This (monotonically increasing) seq would start like this,
I guess:

0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,22,...

Hope this is not old hat,
Best,
É.